Florin DIACU

Department of Mathematics and Statistics
University of Victoria
P.O.Box 3045
Victoria, B.C. 
Canada, V8W 3P4
diacu@math.uvic.ca
 

On the Anisotropic Manev Problem

Abstract : The anisotropic Kepler problem describes the motion of two bodies in an Euclidean space in which the classical gravitational force acts differently in each direction. It has been considered in order to find connections between classical and quantum mechanics.

The Manev problem is isotropic, but the potential is the sum between the inverse and the inverse square of the distance. Here we combine the above two aspects, that is, in the Manev potential we include an anisotropy of the space. Using McGehee coordinates, we blow up the collision singularity, paste a collision manifold to the phase space, study the flow on and near the collision manifold, and find a positive-measure set of collision orbits. In the zero-energy case

we study the escape solutions by defining an infinity manifold. Then we describe all possible connections between equilibria and/or cycles at collision and at infinity. In this way we find the main qualitative features of the global flow on the zero-energy level.